# Constructive Two-View Gauge Initialization - ID: 2048 - Canonical URL: https://synthcore.org/idea/2048/constructive-two-view-gauge-initialization - API JSON: https://synthcore.org/api/idea/2048.json - API Markdown: https://synthcore.org/api/idea/2048.md - Verification status: beats_baseline - Source: [arXiv:2608.09464](https://arxiv.org/abs/2608.09464) - Category: training - Solves: stability, speed, accuracy - ML areas: optimizer, initialization, rnn, ssm, world-model - Math tags: optimization, linear-algebra, geometry, control-theory, dynamical-systems - Ratings: usefulness 7/10; difficulty 4/10; novelty 8/10 ## Idea description Initialize latent coordinate-frame parameters analytically from two temporally separated neural predictions instead of starting joint optimization from arbitrary translation and orientation. This removes the continuous gauge before backpropagation and should prevent EKF-like or gradient-based failures caused by large yaw and position initialization errors. ## Mathematical statement Let d_q=q_2-q_1 and d_ell=ell_2^v-ell_1^v be the global and relay-frame displacement vectors at two views. In the noiseless planar case d_q=R(psi)d_ell, so the relative yaw is psi_hat=atan2(d_{q,y},d_{q,x})-atan2(d_{ell,y},d_{ell,x}) modulo 2pi, provided norm(d_ell)>0. Given this yaw, the relay translation is x_hat=q_1-R(psi_hat)ell_1^v. If the target vector is observed in the relay frame, the target estimate is p_hat=x_hat+R(psi_hat)ell_1^t; for noisy data, average these estimates over all views with robust weights. In a neural setting q_k and ell_k^v may be predicted by a pose encoder and sensor encoder, while ell_k^t is a learned target-bearing representation. The initialization is used only to choose the gauge; subsequent optimization can refine all quantities. ## Key formulas - $$d_q=q_2-q_1,\qquad d_\ell=\ell_2^v-\ell_1^v,\qquad d_q=R(\psi)d_\ell,$$ - $$\hat\psi=\operatorname{atan2}(d_{q,y},d_{q,x})-\operatorname{atan2}(d_{\ell,y},d_{\ell,x})\pmod{2\pi},$$ - $$\hat x=q_1-R(\hat\psi)\ell_1^v,\qquad \hat p=\hat x+R(\hat\psi)\ell_1^t,$$ - $$\min_{x,p,\psi}\sum_{k=1}^{T}\rho\!\left(\|q_k-x-R(\psi)\ell_k^v\|_{\Sigma_v^{-1}}^2+\|p-x-R(\psi)\ell_k^t\|_{\Sigma_t^{-1}}^2\right),$$ ## Implementation notes 1. Integration point: apply this as a warm-start layer before training a neural sensor-fusion model, recurrent filter, differentiable optimizer, or latent-variable encoder. Select two frames from each sequence with the largest predicted displacement norm, rather than using adjacent frames. 2. Pseudocode: encode q[1:T], ell_v[1:T], and ell_t[1:T]; choose (a,b)=argmax norm(q[b]-q[a]); compute dq=q[b]-q[a] and dl=ell_v[b]-ell_v[a]; reject the pair if norm(dl)